A Ferris wheel has a radius of 48.3 meters, and (while spinning) one rotation takes 130 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 389 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=48.3\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{130}\]
\[B~\approx~0.0483322\]
Notice,
\[x~=~48.3\cdot\sin(0.0483322\cdot 389)\]
\[x~=~-5.6978296\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 5.6978296\]
Question
A Ferris wheel has a radius of 68.6 meters, and (while spinning) one rotation takes 147 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 357 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=68.6\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{147}\]
\[B~\approx~0.0427428\]
Notice,
\[x~=~68.6\cdot\sin(0.0427428\cdot 357)\]
\[x~=~28.0373251\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 28.0373251\]
Question
A Ferris wheel has a radius of 80.7 meters, and (while spinning) one rotation takes 378 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 830 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=80.7\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{378}\]
\[B~\approx~0.0166222\]
Notice,
\[x~=~80.7\cdot\sin(0.0166222\cdot 830)\]
\[x~=~76.1544715\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 76.1544715\]
Question
A Ferris wheel has a radius of 25.7 meters, and (while spinning) one rotation takes 93 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 239 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=25.7\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{93}\]
\[B~\approx~0.0675611\]
Notice,
\[x~=~25.7\cdot\sin(0.0675611\cdot 239)\]
\[x~=~-10.6187087\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 10.6187087\]
Question
A Ferris wheel has a radius of 54.3 meters, and (while spinning) one rotation takes 481 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 1116 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=54.3\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{481}\]
\[B~\approx~0.0130628\]
Notice,
\[x~=~54.3\cdot\sin(0.0130628\cdot 1116)\]
\[x~=~48.9069108\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 48.9069108\]
Question
A Ferris wheel has a radius of 69.6 meters, and (while spinning) one rotation takes 409 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 886 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=69.6\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{409}\]
\[B~\approx~0.0153623\]
Notice,
\[x~=~69.6\cdot\sin(0.0153623\cdot 886)\]
\[x~=~60.7104623\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 60.7104623\]
Question
A Ferris wheel has a radius of 124.9 meters, and (while spinning) one rotation takes 467 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 1163 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=124.9\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{467}\]
\[B~\approx~0.0134544\]
Notice,
\[x~=~124.9\cdot\sin(0.0134544\cdot 1163)\]
\[x~=~8.3968309\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 8.3968309\]
Question
A Ferris wheel has a radius of 23.9 meters, and (while spinning) one rotation takes 44 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 102 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=23.9\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
